The expression provided in the image is:

$2\sqrt{2a^3} - 16\sqrt{2b^3} + c^3 + 12abc$

To simplify this expression, let's break it down step by step.

1. Simplifying the Square Root Terms

• For $2\sqrt{2a^3}$:

  • Recognize that $a^3 = a^2 \cdot a$, so $\sqrt{a^3} = \sqrt{a^2 \cdot a} = a\sqrt{a}$.
  • Thus, $2\sqrt{2a^3} = 2\sqrt{2} \cdot a\sqrt{a}$.

• For $-16\sqrt{2b^3}$:

  • Similarly, $\sqrt{b^3} = b\sqrt{b}$.
  • Thus, $-16\sqrt{2b^3} = -16\sqrt{2} \cdot b\sqrt{b}$.

2. Combine All Terms

After simplifying, the expression can be rewritten as:

$2\sqrt{2} \cdot a\sqrt{a} - 16\sqrt{2} \cdot b\sqrt{b} + c^3 + 12abc$

Final Expression

The expression in its simplified form is:

$2\sqrt{2} \cdot a\sqrt{a} - 16\sqrt{2} \cdot b\sqrt{b} + c^3 + 12abc$

Would you like to further explore any part of this expression or have any questions?

Related Questions:

1. How would the expression change if $a = b = c = 1$?
2. Can the expression be factored further?
3. What happens if we substitute $a = 2$, $b = 3$, and $c = 4$?
4. How can we differentiate this expression with respect to $a$?
5. What is the significance of the square root in the context of this expression?

Tip:

When simplifying expressions involving square roots, always look for opportunities to break down the terms inside the square root to see if any factors can be taken outside.